\subsection{Molecule in Fermi Sea}
In sec. \ref{subsec:oneMolInFermiSea}, the close-channel wave function varies with the density (Fermi momentum $k_F$), and in turn the coupling between the channel.  In a very rough sense, the spread of Fermi sea should crowded out the close-channel in low-k or in the large space, and increase its short-range part.  This should enhance the coupling, or $\kappa$ in \cite{Leggett}.  So in pinciple, the desity affects the resonance position in principle.  But there is the some conceptual problem, the $a_s$ is defined in a two-body, i.e., 0 density level, so does the resonance.  On the other hand, T-matrix is well-defined in the many-body context, which should be affected by the change of coupling. All these are in simply normal state, no invoke of the superfluid state.  

\subsection{Tony's comment}
He suggest me to consider the normalization of the new function $\chi_k$ and comparing it to $1/r_0$.  The new normalization also changes the wave-function besides the blocking all the states below Fermi sea.  And he suggest the former is more severe problem.   I feel the compaing with $1/r_0$ is important as the coupling presumably happens within this range.  